数学代写|黎曼几何代写Riemannian geometry代考|MAST90143

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数学代写|黎曼几何代写Riemannian geometry代考|Negative Curvature: The Hyperbolic Plane

The negative constant curvature model is the hyperbolic plane $H_{r}^{2}$ obtained as the surface of $\mathbb{R}^{3}$, endowed with the hyperbolic metric, defined as the zero level set of the function
$$a(x, y, z)=x^{2}+y^{2}-z^{2}+r^{2} .$$
Indeed, this surface is a two-fold hyperboloid, so we can restrict our attention to the set of points $H_{r}^{2}=a^{-1}(0) \cap{z>0}$.

In analogy with the positive constant curvature model (which is the set of points in $\mathbb{R}^{3}$ whose Euclidean norm is constant) the negative constant curvature model can be seen as the set of points whose hyperbolic norm is constant in $\mathbb{R}^{3}$. In other words,
$$H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{2}\right} \cap{z>0}$$
The hyperbolic Gauss map associated with this surface can be easily computed, since it is explicitly given by
$$\mathcal{N}: H_{r}^{2} \rightarrow H^{2}, \quad \mathcal{N}(q)=\frac{1}{r} \nabla_{q} a$$
Exercise 1.63 Prove that the Gaussian curvature of $H_{r}^{2}$ is $\kappa=-1 / r^{2}$ at every point $q \in H_{r}^{2}$.

We can now discuss the structure of geodesics and curves with constant geodesic curvature on the hyperbolic space. We start with a result that can be proved in an analogous way to Proposition $1.60$. The proof is left to the reader.
Proposition 1.64 Let $\gamma:[0, T] \rightarrow H_{r}^{2}$ be a curve with unit speed and constant geodesic curvature equal to $c \in \mathbb{R}$. For every vector $w \in \mathbb{R}^{3}$, the function $\alpha(t)=\langle\dot{\gamma}(t) \mid w\rangle_{h}$ is a solution of the differential equation
$$\ddot{\alpha}(t)+\left(c^{2}-\frac{1}{r^{2}}\right) \alpha(t)=0 .$$

数学代写|黎曼几何代写Riemannian geometry代考|Tangent Vectors and Vector Fields

Let $M$ be a smooth $n$-dimensional manifold and let $\gamma_{1}, \gamma_{2}: I \rightarrow M$ be two smooth curves based at $q=\gamma_{1}(0)=\gamma_{2}(0) \in M$. We say that $\gamma_{1}$ and $\gamma_{2}$ are equivalent if they have the same first-order Taylor polynomial in some (or, equivalently, in every) coordinate chart. This defines an equivalence relation on the space of smooth curves based at $q$.

Definition 2.1 Let $M$ be a smooth $n$-dimensional manifold and let $\gamma: I \rightarrow$ $M$ be a smooth curve such that $\gamma(0)=q \in M$. Its tangent vector at $q=\gamma(0)$, denoted by
$$\left.\frac{d}{d t}\right|_{t=0} \gamma(t) \quad \text { or } \quad \dot{\gamma}(0),$$
is the equivalence class in the space of all smooth curves in $M$ such that $\gamma(0)=$ $q$ (with respect to the equivalence relation defined above).

It is easy to check, using the chain rule, that this definition is well posed (i.e., it does not depend on the representative curve).

Definition $2.2$ Let $M$ be a smooth $n$-dimensional manifold. The tangent space to $M$ at a point $q \in M$ is the set
$$T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \gamma(0)=q\right} .$$ It is a standard fact that $T{q} M$ has a natural structure of an $n$-dimensional vector space, where $n=\operatorname{dim} M$.

Definition 2.3 A smooth vector field on a smooth manifold $M$ is a smooth map
$$X: q \mapsto X(q) \in T_{q} M$$
that associates with every point $q$ in $M$ a tangent vector at $q$. We denote by $\operatorname{Vec}(M)$ the set of smooth vector fields on $M$.

In coordinates we can write $X=\sum_{i=1}^{n} X^{i}(x) \partial / \partial x_{i}$, and the vector field is smooth if its components $X^{i}(x)$ are smooth functions. The value of a vector field $X$ at a point $q$ is denoted, in what follows, by both $X(q)$ and $\left.X\right|_{q}$.

数学代写|黎曼几何代写Riemannian geometry代考|Flow of a Vector Field

Given a complete vector field $X \in \operatorname{Vec}(M)$ we can consider the family of maps
$$\phi_{t}: M \rightarrow M, \quad \phi_{t}(q)=\gamma(t ; q), \quad t \in \mathbb{R}{2}$$ where $\gamma(t ; q)$ is the integral curve of $X$ starting at $q$ when $t=0$. By Theorem $2.5$ it follows that the map $$\phi: \mathbb{R} \times M \rightarrow M{,} \quad \phi(t, q)=\phi_{t}(q)$$
is smooth in both variables and the family $\left{\phi_{t}, t \in \mathbb{R}\right}$ is a one-parametric subgroup of Diff $(M)$; namely, it satisfies the following identities:
\begin{aligned} \phi_{0} &=\mathrm{Id}{+} \ \phi{t} \circ \phi_{s} &=\phi_{s} \circ \phi_{t}=\phi_{t+s}, \quad \forall t, s \subset \mathbb{R}, \ \left(\phi_{t}\right)^{-1} &=\phi_{-t}, \quad \forall t \in \mathbb{R} . \end{aligned}

Moreover, by construction, we have
$$\frac{\partial \phi_{t}(q)}{\partial t}=X\left(\phi_{t}(q)\right), \quad \phi_{0}(q)=q, \quad \forall q \in M$$
The family of maps $\phi_{t}$ defined by $(2.5)$ is called the flow generated by $X$. For the flow $\phi_{t}$ of a vector field $X$ it is convenient to use the exponential notation $\phi_{t}:=e^{t X}$, for every $t \in \mathbb{R}$. Using this notation, the group properties (2.6) take the form
$$\begin{gathered} e^{0 X}=\mathrm{Id}, \quad e^{t X} \circ e^{s X}=e^{s X} \circ e^{t X}=e^{(t+s) X}, \quad\left(e^{t X}\right)^{-1}=e^{-t X} \ \frac{d}{d t} e^{t X}(q)=X\left(e^{t X}(q)\right), \quad \forall q \in M \end{gathered}$$
Remark $2.8$ When $X(x)=A x$ is a linear vector field on $\mathbb{R}^{n}$, where $A$ is an $n \times n$ matrix, the corresponding flow $\phi_{t}$ is the matrix exponential $\phi_{t}(x)=e^{t A} x$.

数学代写|黎曼几何代写Riemannian geometry代考|Negative Curvature: The Hyperbolic Plane

H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{ 2}\right} \cap{z>0}H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{ 2}\right} \cap{z>0}

ñ:Hr2→H2,ñ(q)=1r∇q一个

数学代写|黎曼几何代写Riemannian geometry代考|Tangent Vectors and Vector Fields

dd吨|吨=0C(吨) 或者 C˙(0),

T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \伽马(0)=q\right} 。T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \伽马(0)=q\right} 。一个标准的事实是吨q米有一个自然的结构n维向量空间，其中n=暗淡⁡米.

X:q↦X(q)∈吨q米

数学代写|黎曼几何代写Riemannian geometry代考|Flow of a Vector Field

φ吨:米→米,φ吨(q)=C(吨;q),吨∈R2在哪里C(吨;q)是积分曲线X开始于q什么时候吨=0. 按定理2.5随之而来的是地图

φ:R×米→米,φ(吨,q)=φ吨(q)

φ0=我d+ φ吨∘φs=φs∘φ吨=φ吨+s,∀吨,s⊂R, (φ吨)−1=φ−吨,∀吨∈R.

∂φ吨(q)∂吨=X(φ吨(q)),φ0(q)=q,∀q∈米

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